What is a private browsing app on the iPhone 5s?

That's 2 questions. Welcome to iMore!
1. Private browsing in Safari keeps your visited sites secret with no trail. No history is kept in private browsing. There are other advantages, but that's the main thing.
2. Not enough information to answer that. What exactly is happening with your email? Do you use the stock Mail app? Have you signed in to your account in Settings?